1 Introduction

The impact of various noise sources on the beams in high-energy synchrotrons has been studied extensively in the past. Typically, the focus is on the consequent emittance growth and reduction of luminosity in colliders, or alternatively on how this emittance growth can be suppressed with feedback systems. Recently, it has also been found that the interplay between noise and wakefields can cause instabilities that occur after a latency. Such instabilities have been observed in the Large Hadron Collider (LHC), both in operation and in dedicated experiments [1, 2], and the mechanism behind has been explained semi-analytically [3, 4]. This mechanism is a major concern for the high-luminosity LHC (HL-LHC), which will have beams of higher brightness and also new components such as crab cavities that will introduce a new type of noise. Therefore, we need a better understanding of how the noise excites the beam, including the interplay with other mechanisms. Here, the focus is on the interplay with wakefields.

In this paper, we study the evolution of beam distribution perturbations within the framework of the linear Vlasov equation, as in Ref. [5]. In the absence of wakefields, we use this framework to re-derive the beam transfer function (BTF) [6, 7] including linear chromaticity [8]. The BTF is the amplitude of the transverse betatron oscillation of the beam at a certain frequency, relative to the excitation amplitude at that frequency. When we include the wakefields, the distribution perturbations become wakefield-driven beam eigenmodes. We derive what we call the wakefield beam eigenmode transfer function (MTF), which is the BTF equivalent for each eigenmode separately, also taking into consideration the dependence of the modes on the longitudinal coordinates. Based on the MTFs of all the eigenmodes, we arrive at an expression for the BTF including wakefields. This new theory explains observations made in experiments in the past, highlighting the impact of the wakefields on the BTF. With this understanding of the impact of wakefields, the multi-purpose diagnostic tool that is the BTF can be applied more precisely.

By extending the Vlasov equation to second order, we effectively arrive at a Fokker–Planck equation modeling a diffusion driven by noise and wakefields. It was found in Ref. [3] that the diffusion coefficient related to an almost unstable mode is peaked and narrow in tune space. When there is e.g. a vertical detuning proportional to the vertical action, this diffusion causes a local flattening of the distribution as the particles are evenly distributed over the vertical actions corresponding to the tunes of largest diffusion coefficient. The distribution modification corresponds to a modification of the stability diagram [6], eventually leading to a loss of Landau damping. The MTFs based on the linear Vlasov equation is in this paper used to derive a new expression for this diffusion, requiring less restrictive assumptions than the derivation in Ref. [3]. Furthermore, the Vlasov-based MTFs allow for modeling of the full diffusion driven by noise and wakefields, not only the almost unstable modes.

2 Vlasov equation

This section introduces the Vlasov formalism and notation used throughout this paper. It follows closely the instructive explanation of direct Vlasov solvers in Ref. [5], which itself is based on Ref. [9].

The beam will in general be described by a distribution function \( \varPsi (x,x',y,y',z,\delta ;t) \), and the single-particle dynamics are governed by an effective Hamiltonian \( {\mathcal {H}}(x,x',y,y',z,\delta ;t), \) both dependent on the phase space coordinates \({(x,x')}\) in the horizontal plane, \({(y,y')}\) in the vertical plane, and \({(z,\delta )}\) in the longitudinal plane. The vertical and longitudinal phase space coordinates can be expressed in terms of the corresponding action angle coordinates as

$$\begin{aligned} y&= \sqrt{2J_y\beta _y}\cos (\theta _y),&y'&= \sqrt{\frac{2J_y}{\beta _y}}\sin (\theta _y), \end{aligned}$$
(1)
$$\begin{aligned} z&= \sqrt{2J_z\beta _z}\cos (\phi ),&\delta&\equiv \frac{\varDelta p_z}{p_0}= \sqrt{\frac{2J_z}{\beta _z}}\sin (\phi ) , \end{aligned}$$
(2)

where \( \beta _y \) is an effective vertical beta function, corresponding to \({R/Q_{y0}}\) in Ref. [5], \(\beta _z\) is an effective longitudinal beta function, corresponding to \({\eta _sR/Q_s=\eta _s v/\omega _s}\) in Ref. [5], \({p_0=\gamma m_0 v}\) is the longitudinal momentum of the synchronous particle, R is the average radius of the circular machine, \({Q_{y0}}\) is the unperturbed vertical tune, \({Q_s}\) is the synchrotron tune, \( {\omega _s=\omega _0 Q_s} \) is the synchrotron frequency, \( \omega _0\) is the (angular) rotation frequency of the beam around the machine, \(\eta _s\) is the slippage factor, v is the speed of the synchronous particle, \( \gamma \) is the relativistic factor, and \( m_0 \) is the mass of the synchronous particle. By convention, the transverse angles are written as \( \theta _x \) and \( \theta _y \), while the longitudinal angle is written \( \phi \). The horizontal phase space coordinates \( (x,x') \) can be expressed as in Eq. (1) with all instances of y changed to x.

The evolution of the distribution \(\varPsi \) due to the dynamics described by \({\mathcal {H}}\) is governed by the Vlasov equation

$$\begin{aligned} \frac{{\mathrm{d}}\varPsi }{{\mathrm{d}}t} = \left( \frac{\partial }{\partial t} + {\dot{\theta }}_x\frac{\partial }{\partial \theta _x} + {\dot{J}}_x\frac{\partial }{\partial J_x} + {\dot{\theta }}_y\frac{\partial }{\partial \theta _y} + {\dot{J}}_y\frac{\partial }{\partial J_y} + {\dot{\phi }}\frac{\partial }{\partial \phi } + {\dot{J}}_z\frac{\partial }{\partial J_z} \right) \varPsi = 0, \end{aligned}$$
(3)

where the time evolution for this Hamiltonian system is governed by Hamilton’s equations

$$\begin{aligned} {\dot{J}}_y&\equiv \frac{{\mathrm{d}}J_y}{{\mathrm{d}}t} = \frac{\partial {\mathcal {H}}}{\partial \theta _y},&{\dot{\theta }}_y&\equiv \frac{{\mathrm{d}}\theta _y}{{\mathrm{d}}t} = -\frac{\partial {\mathcal {H}}}{\partial J_y}, \end{aligned}$$
(4)
$$\begin{aligned} {\dot{J}}_z&\equiv \frac{{\mathrm{d}}J_z}{{\mathrm{d}}t} = \frac{\partial {\mathcal {H}}}{\partial \phi },&{\dot{\phi }}&\equiv \frac{{\mathrm{d}}\phi }{{\mathrm{d}}t} = -\frac{\partial {\mathcal {H}}}{\partial J_z}. \end{aligned}$$
(5)

The Vlasov equation is typically solved using perturbation theory, assuming that the Hamiltonian \( {{\mathcal {H}}= {\mathcal {H}}_0 + \varDelta {\mathcal {H}}} \) can be written as the sum of an unperturbed part \({\mathcal {H}}_0\), which here will express the focusing of the particles around the design orbit, and a first-order perturbation \({\varDelta {\mathcal {H}}}\), which expresses the weak forces due to, e.g., wakefields or noise

$$\begin{aligned} {\mathcal {H}}_0&= \omega _0( Q_{x0} + Q'_x\delta ) J_x+ \omega _0( Q_{y0} + Q'_y\delta ) J_y - \omega _s J_z \end{aligned}$$
(6)
$$\begin{aligned}&= \omega _0Q_x J_x + \omega _0Q_y J_y - \omega _s J_z \nonumber ,\\ \varDelta {\mathcal {H}}&= -y \frac{F^\mathrm {coh}_y(z;t)}{p_0} = -\sqrt{2J_y\beta _y}\cos (\theta _y)\frac{F^\mathrm {coh}_y(z;t)}{p_0}, \end{aligned}$$
(7)

where \( Q_{y0} \) and \( {Q'_y} \) are the unperturbed betatron tune and linear chromaticity, respectively, in the vertical plane, and \( Q_{x0} \) and \( {Q'_x} \) are the equivalents in the horizontal plane. Here, it has been expressly stated that the vertical coherent force \( F^\mathrm {coh}_y \) only depends on the longitudinal position and time. Hence, this formalism cannot study effects due to, e.g., beam–beam interactions, but it is sufficient for our study of dipolar wakefields and noise. Note that our derivation considers a vertical coherent force, while all results will focus on noise and wakefields in the horizontal plane.

Within such a perturbation formalism, also the distribution \( {\varPsi =\varPsi _0+\varDelta \varPsi } \) can be treated as the sum of an initial equilibrium part \( {\varPsi _0} \) and a first-order perturbation \( {\varDelta \varPsi } \). The constant equilibrium distribution can be given as

$$\begin{aligned} \varPsi _0&= f_0(J_x,J_y)g_0(J_z) , \end{aligned}$$
(8)

dependent only on the invariants of the unperturbed motion, being the actions. The dependence of \( {\mathcal {H}}_0 \) on \( \phi \) through \( {Q'\delta }\) is negligible [5]. Due to negligible coupling between the transverse planes and the longitudinal plane, the equilibrium distribution is separated in the transverse and longitudinal distribution functions \( f_0 \) and \( g_0 \), respectively. The equilibrium distribution is here normalized such that

$$\begin{aligned} \int _{0}^{2\pi }{\mathrm{d}}{\theta _x}\int _{0}^{\infty }{\mathrm{d}}{J_x} \int _{0}^{2\pi }{\mathrm{d}}{\theta _y}\int _{0}^{\infty }{\mathrm{d}}{J_y} f_0(J_x,J_y)&= 1, \end{aligned}$$
(9)
$$\begin{aligned} \int _{0}^{2\pi }{\mathrm{d}}{\phi }\int _{0}^{\infty }{\mathrm{d}}{J_z} g_0(J_z)&= 1. \end{aligned}$$
(10)

The goal of this derivation is to arrive at an expression for the distribution perturbation \( {\varDelta \varPsi } \). Since \( {\varDelta {\mathcal {H}}} \) depends on y and z, \( \varDelta \varPsi \) depends on \( \theta _y \) and \( \phi \), as well as all the actions, but not \( \theta _x \). The linearized Vlasov equation can now be found by only keeping the terms in Eq. (3) that are linear in the perturbations of the distribution and Hamiltonian as

$$\begin{aligned} \frac{\partial \varDelta \varPsi }{\partial t} - \omega _0Q_y\frac{\partial \varDelta \varPsi }{\partial \theta _y} + \omega _s \frac{\partial \varDelta \varPsi }{\partial \phi } = - \frac{\partial f_0}{\partial J_y}g_0 \sqrt{2J_y\beta _y}\sin (\theta _y) \frac{F^\mathrm {coh}_y}{p_0}, \end{aligned}$$
(11)

where it has been assumed that the derivatives of \( {\varDelta {\mathcal {H}}} \) with respect to the longitudinal action and angle are negligible. Because the coherent force is only in the vertical plane, the horizontal coordinates do not enter at this point. To solve for the distribution perturbation, we first assume that it consists of a single oscillation mode with a complex angular frequency \( \varOmega \) close to \( {\omega _0Q_{y0}} \). By making a Fourier expansion in \( \phi \), it can be shown that the distribution perturbation is given by (Eq. (68) in Ref. [5])

$$\begin{aligned} \varDelta \varPsi = \mathrm {e}^{i\varOmega t} \frac{\partial f_0}{\partial J_y} \sqrt{2J_y \beta _y}\mathrm {e}^{i\theta _y} \mathrm {e}^{-\frac{iQ'_y z}{Q_s \beta _z}} \sum \limits _{l=-\infty }^{\infty } R_l(J_z) \mathrm {e}^{-il \phi }, \end{aligned}$$
(12)

where \( {Q'_yz/Q_s \beta _z} \) is the headtail phase factor due to chromaticity. The transverse part of the distribution perturbation is fully specified. The longitudinal distribution perturbation consist of the Fourier sum over the so far unspecified functions \( R_l(J_z) \), which can be found by solving the linearized Vlasov equation that now takes form (Eq. (69) in Ref. [5])

$$\begin{aligned} \sum \limits _{l=-\infty }^{\infty } R_l(J_z) \mathrm {e}^{-il\phi } (\varOmega -\omega _0 Q_{y0}-l\omega _s) = \mathrm {e}^{-i\varOmega t} \mathrm {e}^{\frac{iQ'_y z}{Q_s \beta _z}} g_0(J_z)\frac{F^\mathrm {coh}_y}{2p_0}. \end{aligned}$$
(13)

In the following, we will interchangeably use the radial coordinate \( {r_z=\sqrt{2J_z\beta _z}}\) to simplify the notation and analytical calculations. This change of coordinate is not used when expressing Hamilton’s equations; hence, it does not need to be a canonical transformation [10].

3 Beam transfer function

A transfer function of a system represents the relationship between the output signal and the input signal. The transverse beam transfer function (BTF) typically represents the relationship between the transverse oscillation amplitude of the center of mass of the beam (output) and the excitation signal provided by magnets (input) [6,7,8]. The excitation signal can either be broadband or consists of a single frequency.

3.1 Beam transfer function with chromaticity

First, we want to derive an expression for the BTF including chromaticity. We consider the coherent force \( F^\mathrm {coh}_y \) to be an external harmonic driving force of amplitude \( A_y \) and frequency \( \varOmega \)

$$\begin{aligned} F^\mathrm {coh}_y(z,t) = A_y\mathrm {e}^{i\varOmega t}. \end{aligned}$$
(14)

Putting it in Eq. (13) and using the Jacobi–Anger expansion (Eq. (8.511.4) in Ref. [11]) give

$$\begin{aligned} \sum \limits _{l=-\infty }^{\infty } R_l(r_z)\mathrm {e}^{-il\phi }(\varOmega -\omega _0Q_{y\,0}-l\omega _s) = \frac{A_y}{2p_0} g_0(r_z) \sum \limits _{l=-\infty }^{\infty } i^l \mathrm {e}^{il\phi } J_l \left( \frac{Q'_yr_z}{Q_s \beta _z} \right) , \end{aligned}$$
(15)

where \( J_l(\cdot ) \) are the Bessel functions of order l. By equating terms with \(\mathrm {e}^{-il\phi }\), one gets an expression for the longitudinal modes

$$\begin{aligned} R_l(r_z)\mathrm {e}^{-il\phi }=\frac{A_y}{2p_0}\frac{g_0(r_z)}{\varOmega -\omega _0Q_{y\,0}-l\omega _s} i^{-l}\mathrm {e}^{-il\phi }J_{-l}\left( \frac{Q'_yr_z}{Q_s \beta _z}\right) . \end{aligned}$$
(16)

Inserting this into the expression for the distribution perturbation in Eq. (12) gives

$$\begin{aligned} \varDelta \varPsi = \frac{A_y}{2p_0}\mathrm {e}^{i\varOmega t} \mathrm {e}^{i\theta _y} \frac{\partial f_0}{\partial J_y} g_0(r_z) \sqrt{2J_y\beta _y} \sum \limits _{k,l}\frac{i^{l-k}\mathrm {e}^{i(k-l)\phi }}{\varOmega -\omega _0Q_{y}(J_x,J_y)-l\omega _s}J_{k}\left( \frac{Q'_yr_z}{Q_s \beta _z}\right) J_{l}\left( \frac{Q'_yr_z}{Q_s \beta _z}\right) , \end{aligned}$$
(17)

where we have introduced a weak transverse detuning that is a function of the transverse actions as in Ref. [6], such as the detuning driven by octupole magnets [12]

$$\begin{aligned} Q_y(J_x,J_y) = Q_{0y} + a_y J_y + b_y J_x. \end{aligned}$$
(18)

We are now in a position to calculate the BTF, defined as the beam response to the driving force. By realizing that only terms with \( {k=l} \) do not vanish when integrating \( \varDelta \varPsi \) over \(\phi \) and that the equilibrium distribution is normalized as in Eqs. (9) and (10), one finds

$$\begin{aligned} \begin{aligned} \mathrm {BTF}&\equiv \frac{y^\mathrm {dip}}{F^\mathrm {coh}_y}\frac{p_0}{\beta _y} \equiv \frac{p_0}{F^\mathrm {coh}_y\beta _y}\frac{ \int _{0}^{2\pi }{\mathrm{d}}{\theta _x}\int _{0}^{\infty }{\mathrm{d}}{J_x} \int _{0}^{2\pi }{\mathrm{d}}{\theta _y}\int _{0}^{\infty }{\mathrm{d}}{J_y} \int _{0}^{2\pi }{\mathrm{d}}{\phi }\int _{0}^{\infty }{\mathrm{d}}{J_z} (\varPsi _0 + \varDelta \varPsi )\sqrt{2J_y\beta _y}\cos (\theta _y) }{ \int _{0}^{2\pi }{\mathrm{d}}{\theta _x}\int _{0}^{\infty }{\mathrm{d}}{J_x} \int _{0}^{2\pi }{\mathrm{d}}{\theta _y}\int _{0}^{\infty }{\mathrm{d}}{J_y} \int _{0}^{2\pi }{\mathrm{d}}{\phi }\int _{0}^{\infty }{\mathrm{d}}{J_z} (\varPsi _0 + \varDelta \varPsi ) }\\&= 4\pi ^3 \sum \limits _{l=-\infty }^{\infty } \int _0^\infty \int _0^\infty \frac{J_y \displaystyle \frac{\partial f_0}{\partial J_y}}{\varOmega -\omega _0Q_{y}(J_x,J_y)-l\omega _s}{\mathrm{d}}{J_x}{\mathrm{d}}{J_y} \int _0^\infty J_{l}\left( \frac{Q'_yr_z}{Q_s \beta _z}\right) ^2 g_0(r_z) \frac{r_z{\mathrm{d}}{r_z}}{\beta _z}, \end{aligned} \end{aligned}$$
(19)

where \( y^\mathrm {dip}\) is the dipolar moment of the bunch. The dipolar moment is scaled by the beta function \( \beta _y \) so that it is given in terms of \( y^\prime \), as seen in Eq. (1). The force is scaled by the momentum \( p_0 \), as in Eq. (34) in Ref. [5], to be the change of \( y^\prime \). The BTF can be put in the form

$$\begin{aligned} \mathrm {BTF}= \sum \limits _l {\mathcal {T}}_{l}(\varOmega ) w_{0l}(Q'_y), \end{aligned}$$
(20)

with the familiar dispersion integral [6] and a weight that depends on chromaticity

$$\begin{aligned} {\mathcal {T}}_{l}(\varOmega )&\equiv 2\pi ^2 \int _0^\infty \int _0^\infty \frac{J_y \displaystyle \frac{\partial f_0}{\partial J_y}}{\varOmega -\omega _0Q_{y}(J_x,J_y)-l\omega _s}{\mathrm{d}}{J_x}{\mathrm{d}}{J_y} , \end{aligned}$$
(21)
$$\begin{aligned} w_{0l}(Q'_y)&\equiv 2\pi \int _0^\infty J_{l}\left( \frac{Q'_yr_z}{Q_s \beta _z}\right) ^2 g_0(r_z)\frac{r_z{\mathrm{d}}{r_z}}{\beta _z}. \end{aligned}$$
(22)

In the absence of chromaticity, \( {w_{0l}} \) is 1 for \( {l=0} \) and 0 otherwise, returning \( {\mathcal {T}}_0 \) as the \( \mathrm {BTF}\). For a Gaussian distribution with variation \( \sigma _z^2=\langle J_z\beta _z\rangle \) and use of Eq. (6.633) in Ref. [11], we recover the result in Ref. [8]

$$\begin{aligned} g_0(r_z)&=\frac{\beta _z}{2\pi \sigma _z^2}\mathrm {e}^{-\frac{r_z^2}{2\sigma _z^2}},&w^G_{0l}(Q'_y)&= \mathrm {e}^{-\left( \frac{Q'_y\sigma _z}{Q_s \beta _z}\right) ^2}I_l\left( \left( \frac{Q'_y\sigma _z}{Q_s \beta _z}\right) ^2\right) , \end{aligned}$$
(23)

where \( I_l(\cdot ) \) are the modified Bessel functions of the first kind.

3.2 Wakefield beam eigenmode transfer function

In Sect. 3.1, we considered the coherent driving force in Eq. (14) with beam-independent amplitude. Now, we will study the driving by the coherent wake force \( F^\mathrm {wake}_y \), which depends on the transverse beam motion. In the following, we adopt the weak wakefield approximation, assuming that the beam modes of different azimuthal number l are linearly independent. To distinguish between different modes, we introduce the subscript m, such that mode m has azimuthal number \( l_m \) and so forth. Note that different modes \( {m\ne n} \) can have the same azimuthal mode number \( {l_m=l_n} \), but a particular mode has only a single \( l_m \). For ease of notation, the longitudinal polar radius \( {r_z=\sqrt{2J_z\beta _z}} \) is used instead of the longitudinal action.

The wake force due to the passings of a beam eigenmode at all turns k can be given as (Eq. (85) in Ref. [5])

$$\begin{aligned} F^\mathrm {wake}_{y\,m}(z;t)&= \mathrm {e}^{i\varOmega _m t}\frac{i^{1+l_m}N e^2 \omega _0}{2\pi \beta _z Q_{y0} } \sum \limits _{k=-\infty }^\infty Z_y(\varOmega _m+k\omega _0)\mathrm {e}^{-i(\varOmega _m+k\omega _0)\frac{z}{v}} {\mathcal {I}}_{l_m k}, \end{aligned}$$
(24)
$$\begin{aligned} {\mathcal {I}}_{l_m k}&= \int _{0}^{\infty }R_{l_m}(r_z')J_{l_m}\left[ \left( \varOmega _m+k\omega _0 - \frac{Q'_y v}{Q_s \beta _z}\right) \frac{r_z'}{v}\right] r_z'{\mathrm{d}}{r_z'}, \end{aligned}$$
(25)

where \( Z_y \) is the impedance function, i.e. the Fourier transform of the wake function, and N is the number of particles in the bunch. With this expression for the coherent force, Sacherer’s integral equation [13] takes the form (Eq. (88) in Ref. [5])

$$\begin{aligned}&R_{l_m}(r_z) (\varOmega _m -\omega _0 Q_{y0}-l_m\omega _s) \nonumber \\&\quad = g_0(r_z) \frac{iN e^2 \omega _0}{4\pi \beta _z Q_{y0} p_0} \sum \limits _{k=-\infty }^\infty Z_y(\varOmega _m+k\omega _0) J_{l_m}\left[ \left( \varOmega _m+k\omega _0 - \frac{Q'_y v}{Q_s \beta _z}\right) \frac{r_z}{v}\right] {\mathcal {I}}_{l_m k}, \end{aligned}$$
(26)

where it has been used that the modes have a single azimuthal mode number \( l_m \). This set of equations can be solved numerically with direct Vlasov solvers, such as DELPHI [14], to find the transverse eigenmodes \( {\varDelta \varPsi _m} \) and their frequencies \( \varOmega _m \). These mode details can also be found using codes that implement the circulant matrix model, such as BimBim [15]. Both these codes have been used in this paper.

The wake force due to a dipolar eigenmode is often expressed as proportional to the dipolar transverse offset, \(F^\mathrm {wake}_y/p_0=-2\varDelta \varOmega \langle y\rangle /\beta _y \) [6], where \( \varDelta \varOmega \) is the tune shift driven by the wakefields. The factor \( \beta _y/p_0 \) occurs here as in Eq. (19) to properly scale the force to the change of position, as in Ref. [5]. Here, we aim at generalizing this proportionality of the wake force for an eigenmode with transverse offset dependent on the longitudinal phase space. To that end, we calculate the transverse offset of the mode as a function of the longitudinal coordinates

$$\begin{aligned} \begin{aligned} \langle y(r_z,\phi ;t)\rangle _m&= \frac{ \int _{0}^{2\pi }{\mathrm{d}}{\theta _x}\int _{0}^{\infty }{\mathrm{d}}{J_x} \int _{0}^{2\pi }{\mathrm{d}}{\theta _y}\int _{0}^{\infty }{\mathrm{d}}{J_y} \sqrt{2J_y\beta _y}\cos (\theta _y) (\varPsi _0 + \varDelta \varPsi _m) }{ \int _{0}^{2\pi }{\mathrm{d}}{\theta _x}\int _{0}^{\infty }{\mathrm{d}}{J_x} \int _{0}^{2\pi }{\mathrm{d}}{\theta _y}\int _{0}^{\infty }{\mathrm{d}}{J_y} (\varPsi _0 + \varDelta \varPsi _m) }\\&= - \frac{\beta _y}{g_0(r_z)} \mathrm {e}^{i\varOmega _m t} \mathrm {e}^{-\frac{iQ'_y z}{Q_s \beta _z}} R_{l_m}(r_z) \mathrm {e}^{-il_m \phi }, \end{aligned} \end{aligned}$$
(27)

where it has been used that the average offset due to the equilibrium distribution \( \varPsi _0 \) in the numerator is 0 and the integral over the wakefield eigenmode \( {\varDelta \varPsi _m} \) in the denominator is 0.

In general, there does not exist a scalar proportionality constant between \( {F^\mathrm {wake}_{y\,m}(z;t)}\) and \({\langle y(r_z,\phi ;t)\rangle _m} \) at any given instant of time, since the former depends only on z, while the latter depends on both \( r_z \) and \( \phi \). However, by combining Eqs. (24)–(27), one can find that

$$\begin{aligned} \frac{\beta _y}{p_0}\left\langle {\frac{F^\mathrm {wake}_{y\,m}(z;t)}{\langle y(r_z,\phi ;t)\rangle _m}}\right\rangle _t = -2(\varOmega _m -\omega _0 Q_{y0}-l_m\omega _s) \equiv -2\varDelta \varOmega _m, \end{aligned}$$
(28)

where the time average of the ratio is taken over one synchrotron period \( {\tau _s=1/\omega _s} \). This time averaging corresponds to an averaging over the longitudinal phase \( \phi \), which also enters in z as given by Eq. (2). Therefore, we suggest instead an effective force, as in Ref. [3], that is proportional to the transverse offset, including its dependence on the longitudinal coordinates

$$\begin{aligned} F^{\mathrm {wake}\,\mathrm {eff}}_{y\, m}(r_z,\phi ;t) = -\frac{2 p_0}{\beta _y}\varDelta \varOmega _m \langle y(r_z,\phi ;t)\rangle _m. \end{aligned}$$
(29)

A noise force acting on the beam is now introduced in addition to the wake force

$$\begin{aligned} F^\mathrm {noise}_y = \frac{p_0}{\beta _y}\xi (z;t). \end{aligned}$$
(30)

Since the noise does not depend on the beam, it is not expected to change the shape of the wakefield beam eigenmodes \( \varDelta \varPsi _m \). However, it will excite more than one mode at once. Hence, we can no longer consider a single wakefield beam eigenmode. Instead, we get

$$\begin{aligned} \varDelta \varPsi&= \varDelta \varPsi _m\rightarrow & {} \varDelta \varPsi&= \sum _m \varDelta \varPsi _m, \end{aligned}$$
(31)
$$\begin{aligned} \langle y(r_z,\phi ;t)\rangle&= \langle y(r_z,\phi ;t)\rangle _m\rightarrow & {} \langle y(r_z,\phi ;t)\rangle&= \sum _m \langle y(r_z,\phi ;t)\rangle _m \equiv \sum _m \chi _m(t) m_m(r_z,\phi ), \end{aligned}$$
(32)

where we have introduced as in Ref. [3] the time-dependent mode amplitude \( \chi _m(t) \) and normalized constant eigenmode shape functions \( m_m(r_z,\phi ) \), which by comparison to Eq. (27) can be found to be

$$\begin{aligned} m_m(r_z,\phi ) = A_m \mathrm {e}^{-\frac{iQ'_y z}{Q_s \beta _z}} \frac{R_{l_m}(r_z)}{g_0(r_z)} \mathrm {e}^{-il_m \phi }, \end{aligned}$$
(33)

where \( A_m \) is a normalization constant such that

$$\begin{aligned} \langle m_m^*m_m\rangle _z \equiv \int _{0}^{2\pi } {\mathrm{d}}{\phi } \int _{0}^{\infty } {\mathrm{d}}{J_z} m_m^*m_m g_0 = 1, \end{aligned}$$
(34)

where the superscripted \( *\) implies a complex conjugation and it has been used that \( g_0 \) is normalized, as given by Eq. (10). Note that the average over the longitudinal coordinates, denoted by \( \langle \cdot \rangle _z \), includes the unperturbed longitudinal distribution \( g_0 \) in the integrand. These excited modes are not necessarily orthogonal, but are still assumed to be linearly independent and can therefore be treated independently.

The noise can be modeled as a single kick per particle per turn. The noise signal \( \xi (z;t) \) can be decomposed in orthonormal noise shape functions \( \varXi _i(z) \), which are polynomials of order i, being \( {\varXi _0=1} \), \( {\varXi _1=z/\sigma _z} \), and higher-order polynomials for a longitudinally Gaussian distribution. These noise shape functions can themselves be decomposed in the eigenmode shape functions \( m_m \)

$$\begin{aligned} \begin{aligned} \xi (z;t)&= \sum _i \xi _i(t) \varXi _i(z) = \sum _i \xi _i(t) \sum _m \eta _{i\,m} m_m(r_z,\phi ), \end{aligned} \end{aligned}$$
(35)

where the noise signals \( \xi _i(t) \) are assumed to only have power in positive frequencies \( {\omega \in [0,\omega _0/2)} \) and the noise-mode moments \( \eta _{i\,m} \) are the projection coefficients of the noise shape function \( \varXi _i \) on the mode shape function \( m_m \). The noise shape functions are normalized over the initial equilibrium distribution function as

$$\begin{aligned} \langle \varXi _i^*\varXi _j\rangle _z = \sum _m \eta _{i\,m}^*\langle m_m^*\varXi _j \rangle _z \equiv \sum _m \eta _{i\,m}^*{\tilde{\eta }}_{j\,m} = \delta _{i,j}, \end{aligned}$$
(36)

where \( \delta _{i,j} \) is the Kronecker delta and \( {{\tilde{\eta }}_{j\,m}\equiv \langle m_m^*\varXi _j \rangle _z} \) is equal to \( \eta _{j\,m} \) if the modes are orthogonal. If not, \( \eta _{j\,m} \) can be found by solving a linear system of equations. Most noise sources produce noise that is constant across a bunch, being proportional to the constant \( {\varXi _0(z)=1} \). This type of noise will be referred to as rigid or dipolar noise. The shape of crab cavity amplitude noise, on the other hand, is proportional to the longitudinal position, \( {\varXi _1(z)\propto z} \). To the authors’ knowledge, there are no significant noise sources for which higher-order terms in z are necessary.

The full coherent force can now be decomposed as:

$$\begin{aligned} \begin{aligned} F^\mathrm {coh}_y&= F^{\mathrm {wake}\,\mathrm {eff}}_y + F^\mathrm {noise}_y\\&= \frac{p_0}{\beta _y}\sum _m \left( -2\varDelta \varOmega _m\chi _m(t) + \sum _i\xi _i(t) \eta _{i\,m} \right) m_m(r_z,\phi )\\&\equiv \sum _m F^\mathrm {coh}_{y\,m}, \end{aligned} \end{aligned}$$
(37)

where the coherent force \( F^\mathrm {coh}_{y\,m} \) proportional to mode m has been defined.

To get the transfer function of mode m, one must now calculate the expected excitation due to noise at a given driving frequency \( \varOmega \). The mode is then forced to oscillate at the driving frequency, even if it is different from the natural eigenfrequency of a mode \( \varOmega _m \). In general, the closer the driving frequency is to the eigenfrequency, the larger the amplitude of the modes motion is expected to be. By combining Eqs. (12) and (13) with the assumption of a single azimuthal mode number \( l_m \) per linearly independent mode, meaning that the sums over l are exchanged for a single term, one gets the expression

$$\begin{aligned} \varDelta \varPsi _m = \frac{\sqrt{2 J_y \beta _y} \mathrm {e}^{i\theta _y} \displaystyle \frac{\partial \varPsi _0}{\partial J_y}}{ \varOmega -\omega _0 Q_y(J_x,J_y)-l_m\omega _s} \frac{F^\mathrm {coh}_{y\,m}}{2p_0}, \end{aligned}$$
(38)

where we have again introduced a weak transverse detuning. The average offset of mode m at frequency \( \varOmega \) can be calculated with the first line of Eqs. (27) and (38)

$$\begin{aligned} \begin{aligned} \langle y(r_z,\phi ;t)\rangle _m&= \chi _m(t) m_m(r_z,\phi ) \\&= {\mathcal {T}}_{l_m}(\varOmega ) \left( -2\varDelta \varOmega _m\chi _m(t) + \sum _i\xi _i(t) \eta _{i\,m} \right) m_m(r_z,\phi ), \end{aligned} \end{aligned}$$
(39)

where the dispersion integral \( {\mathcal {T}}_{l_m}(\varOmega ) \) in Eq. (21) has reappeared. By rearranging the terms, we get an expression for the wakefield beam eigenmode transfer function (MTF) due to the excitation by noise at frequency \( \varOmega \)

$$\begin{aligned} \frac{{\hat{\chi }}_m(\varOmega )}{\sum _i {\hat{\xi }}_i(\varOmega )\eta _{i\,m}} = \frac{{\mathcal {T}}_{l_m}(\varOmega )}{1 + 2\varDelta \varOmega _m{\mathcal {T}}_{l_m}(\varOmega ) } = \frac{1/2}{\varDelta \varOmega _m-\varDelta \varOmega ^\mathrm {SD}_{l_m}(\varOmega ) } \equiv {\mathcal {T}}^\mathrm {wake}_{l_m}(\varOmega ), \end{aligned}$$
(40)

where the notation \( {{\hat{f}}(\varOmega ) \equiv \int ^\infty _{-\infty } f(t)\exp (-i\varOmega t){\mathrm{d}}{t}} \) indicates the Fourier transform of f(t) . The amplitude of mode m diverges when \( {\varDelta \varOmega _m \rightarrow -1/[2{\mathcal {T}}_{l_m}(\varOmega )]=\varDelta \varOmega ^\mathrm {SD}_{l_m}(\varOmega )} \). \( {\varDelta \varOmega ^\mathrm {SD}_{l_m}(\varOmega )} \) is an important quantity in beam stability, often referred to as the stability diagram. Examples of the stability diagram are illustrated in Fig. 6. A mode is considered to be stabilized by Landau damping when its complex tune is inside (below) the stability diagram [6]. Note also the appearance of \( \eta _{i\,m} \) in the summand in the denominator on the left-hand side. A smaller value of \( \eta _{i\,m} \) means that the noise \( \xi _i \) is less effective at exciting mode m. The combined coherent force from the noise and wakefields related to mode m in Eq. (37) can with this MTF be expressed as:

$$\begin{aligned} {\hat{F}}^\mathrm {coh}_{y\,m} = \frac{p_0}{\beta _y} \frac{ \sum _i {\hat{\xi }}_i(\varOmega )\eta _{i\,m} }{1 + 2\varDelta \varOmega _m{\mathcal {T}}_{l_m}(\varOmega ) } m_m(r_z,\phi ). \end{aligned}$$
(41)

3.3 Beam transfer function including wakefields

The MTF for a single linearly independent beam mode with a single azimuthal mode number \( l_m \) is given in Eq. (40). To express the measurable dipolar BTF, the MTF of all modes must be combined. Since the BTF typically is measured by controlled excitation by a dipolar noise source, which excites all particles in the bunch equally, we will in this section assume that \( \xi _0(t) \) is the only noise signal of nonzero amplitude.

The dipolar mode moment can be calculated as

$$\begin{aligned} y^\mathrm {dip}_m(t) = \chi _m\langle m_m\rangle _z = \chi _m\langle \varXi _0^*m_m\rangle _z= \chi _m {\tilde{\eta }}_{0\,m}^*, \end{aligned}$$
(42)

where it was used that \( {\varXi _0(z)=1} \). By combining with Eq. (40), and assuming only dipolar noise, we get the desired expression for the BTF including wakefields

$$\begin{aligned} \mathrm {BTF}= \frac{{\hat{y}}^\mathrm {dip}}{{\hat{F}}^\mathrm {noise}_y}\frac{p_0}{\beta _y} = \frac{{\hat{y}}^\mathrm {dip}(\varOmega )}{{\hat{\xi }}_0(\varOmega )} = \frac{\sum _m {\hat{y}}^\mathrm {dip}_m(\varOmega )}{{\hat{\xi }}_0(\varOmega )} = \sum _m {\mathcal {T}}^\mathrm {wake}_{l_m}(\varOmega ) {\tilde{\eta }}_{0\,m}^*\eta _{0\,m}, \end{aligned}$$
(43)

where \( F_y^\mathrm {noise}\) is defined in Eq. (30) and \( {\mathcal {T}}^\mathrm {wake}_{l_m}(\varOmega ) \) is defined in Eq. (40). In the absence of chromaticity and wakefields, Eq. (43) returns the standard BTF, \( {{\mathcal {T}}_0(\varOmega )} \). In the limit of negligible wakefields, but still including chromaticity, such that \( {{\mathcal {T}}^\mathrm {wake}_{l_m}={\mathcal {T}}_{l_m}} \), we can by comparison to Eq. (20) calculate the weight of sideband l as

$$\begin{aligned} w^\mathrm {MTF}_{0l} = \sum _m {\tilde{\eta }}_{0\,m}^*\eta _{0\,m} \delta _{l,l_m}. \end{aligned}$$
(44)

The weight \( w^\mathrm {MTF}_{0l} \) is, in the limit of negligible wakefields, equivalent to the weight \( w_{0l} \) in Eq. (22), which was derived in the presence of chromaticity but completely without wakefields.

So far, the derivation has not made any assumptions as to the orthogonality of the modes, leading to the distinction between \( {\tilde{\eta }} \) and \( \eta \) in Eq. (44). If the modes are orthogonal, one has \( {{\tilde{\eta }}_{0\,m}^*\eta _{0\,m}= \left| \eta _{0\,m}\right| ^2}\). This has been found to be true within the numerical accuracy of BimBim for the LHC.

The weights \( w_{0l} \) in Eq. (22) and the weights \( w^\mathrm {MTF}_{0l} \) in Eq. (44), assuming orthogonal modes and using negligible wakefields in DELPHI, are compared in Fig. 1a. This figure shows how the dipolar moment of modes with nonzero azimuthal mode number \( {l_m\ne 0} \) increases with the chromaticity, a behavior that is important for the mechanisms in this paper. How the weights are calculated in DELPHI is explained in Appendix A.

One can alternatively imagine a headtail BTF (\( \mathrm {BTF}^\mathrm {ht}\)), excited by headtail noise \( {\propto \varXi _1} \) whereupon the headtail motion is measured. This is further explained and derived in Appendix B. The theoretical weights \( w_{1l} \) in Eq. (B.7) are in Fig. 1b compared to the weights \( w^\mathrm {MTF}_{1l} \) in Eq. (B.10), assuming orthogonal modes. This figure shows how crab cavity amplitude noise can efficiently excite modes of azimuthal mode number \( {l_m=\pm 1} \) at small chromaticities.

Fig. 1
figure 1

Weights of low-order sidebands of the BTF. In (a) rigid motion due to rigid noise \( {\propto \varXi _0} \). In (b) headtail motion due to headtail noise \( {\propto \varXi _1} \). The fully analytical theories (lines) in Eqs. (22) and (B.7) are compared to the MTF theory (points) in Eqs. (44) and (B.10), based on DELPHI calculations and assuming orthonormal modes

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3.4 Comparison to multi-particle tracking simulations

The expressions derived for the BTF in Eqs. (20) and (43) will here be benchmarked against multi-particle tracking simulations run with COMBI [16, 17]. Calculations made with BimBim, using the impedance model for the LHC at top energy in 2018, have been used to get the mode details required by Eq. (43), i.e. the noise-mode-moments \( \eta _{0m} \) and tune shifts \( \varDelta Q^\mathrm {coh}_m \). The bunch intensity, synchrotron tune, and root-mean-squared (rms) particle momentum spread were the same as given by Table 1, see below. The tracking simulations were run with \( 10^7 \) macro-particles for \( 10^{5}\) turns, from which the BTF was found by calculating the power spectral density of the beam signal and noise using the Welch algorithm. The horizontal noise used was approximately white with an rms kick per turn of \(5 \times 10^{-4}\sigma _{x}\), in units of the beam size.

The new theories and the tracking simulations, both with and without wakefields, show excellent agreement in Fig. 2. It is clearly shown that it is the wakefields in the LHC that cause the shift of the main peak (and side peaks as well) of the BTF, as has been hypothesized based on measurements in the LHC (Fig. 7 in Ref. [7]). Although it is not shown here visually, the theory in Eq. (20) is also able to explain the appearance of loops at the sidebands in the stability diagram reconstructed based on the BTF [18].

Fig. 2
figure 2

BTF in the horizontal plane without chromaticity in (a) and with a linear chromaticity of \({Q'_x=15} \) in (b). In both cases, multi-particle simulations run with COMBI are compared to the BTF excluding wakefields in Eq. (20) and the BTF including wakefields in Eq. (43). The BTFs including wakefields were calculated based on BimBim calculations

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4 Loss of Landau damping driven by diffusion

Instabilities have been observed in the LHC that occur after a long latency, in the order of tens of minutes, after reaching top energy. The mechanism behind these instabilities has just recently been understood [3]: the interplay between noise and wakefields drives a diffusion that eventually renders the beam unstable. In Ref. [3], the expression for the diffusion was based on the MTF for a mode modeled as a single underdamped stochastic harmonic oscillator (USHO). In this section, we will derive the diffusion by use of the MTF in Eq. (40), which is based on the linear Vlasov theory. The new derivation was inspired by the second part of Ref. [19], yet relaxing strong assumptions that prevented quantitative estimates for realistic machines. Most important was to relax the rigid bunch assumption and the simplified wakefield model, which corresponded to only considering mode shape functions \( m_m \) that were independent of the longitudinal coordinates \( r_z \) and \( \phi \). In addition, we have extended from using a one-dimensional tune spread, setting \( b_y \) in Eq. (18) to 0, to using a two-dimensional tune-spread. The new Vlasov theory will be compared to the existing USHO theory.

4.1 Diffusion driven by wakefield eigenmodes

If the vertical noise and wakefields are sufficiently weak to be modeled as a perturbation, they drive a second-order diffusion of the equilibrium distribution. This requires a limited change of the distribution over the correlation period of the coherent force. The diffusion in the vertical plane can be modeled analytically by [20, 21]

$$\begin{aligned} \frac{\partial \varPsi _\mathrm {eq}}{\partial t}&= \frac{\partial }{\partial J_y}\left[ J_y D_y \frac{\partial \varPsi _\mathrm {eq}}{\partial J_y}\right] , \end{aligned}$$
(45)
$$\begin{aligned} D_y&= \lim \limits _{{T} \rightarrow \infty } \frac{1}{{T}} \int \limits _{t_0}^{t_0+{T}}{\mathrm{d}}{t} \int \limits _{t_0}^{t_0+{T}}{\mathrm{d}}{s} \frac{\left\langle {\dot{J}}_y(t) {{\dot{J}}_y(s)}\right\rangle _{z}}{2J_y}, \end{aligned}$$
(46)

where the equilibrium distribution has been renamed \( {\varPsi _0\rightarrow \varPsi _\mathrm {eq}} \) since it will evolve with time. The autocorrelation of the change of action is averaged over the longitudinal coordinates, as how the transverse diffusion depends on the longitudinal action is irrelevant. An equivalent horizontal diffusion term can be added to Eq. (45) to model the diffusion driven by horizontal noise and wakefields. There is assumed no coupling between the two transverse planes in terms of diffusion.

Since the wake force in Eq. (24) is complex, it will give a complex expression for \( {\dot{J}}_y \) by use of Eq. (4). Therefore, we calculate a real expression for the change of action as

$$\begin{aligned} {\dot{J}}_y(t) = 2\mathrm {Re}\left\{ \frac{\partial \varDelta {\mathcal {H}}}{\partial \theta _y}\right\} = -\frac{\sqrt{2J_y\beta _y}}{p_0}\mathrm {Re}\left\{ i\left( \mathrm {e}^{i\theta _y(t)} - \mathrm {e}^{-i\theta _y(t)}\right) \sum _m F^\mathrm {coh}_{y\,m}(r_z,\phi +\omega _s t;t)\right\} , \end{aligned}$$
(47)

where \( \theta _y \) is the angle of the single particle. It can be shown as in Ref. [3] that the vertical single particle angle is

$$\begin{aligned} \theta _y = \theta _{0y} - \omega _0Q_y t + \frac{Q_y'z}{Q_s\beta _z}. \end{aligned}$$
(48)

Since \( F^\mathrm {wake}_{y\,m}\propto \exp (i\varOmega t) \), with \( \varOmega \sim \varOmega _m \), and \( F^\mathrm {noise}\) is assumed to only have power in positive frequencies, only the close-to-resonant term \( {\exp (i\theta _y)\propto \exp (-i\omega _0Q_y t)} \) will give a nonnegligible integrated change of action. The headtail phase factor \( {Q'_yz/Q_s\beta _z} \) in the single particle angle \( \theta _y \) in Eq. (47) will then cancel the headtail phase factor of the coherent force in Eq. (37), contained within the functions \( m_m \) in (33). Note that as we are no longer focused on the behavior of the modes, but rather on their consequences on the individual particles, we are now referring to \( {\omega _y\sim \omega _0 Q_{y0}} \), the frequencies of the individual particles, instead of \( {\varOmega _{m}\sim \omega _0 Q_{y0}+l_m \omega _s} \), the frequency of the mode.

The integrand in Eq. (46), neglecting the averaging over z, becomes

$$\begin{aligned} \frac{{\dot{J}}_y(t) {{\dot{J}}_y(s)}}{2J_y}&\quad = \frac{-\beta _y}{4 p_0^2} \left[ \mathrm {e}^{ i(\theta _{0y} - \omega _y t)} \sum _m F^\mathrm {coh}_{y\,m,z=0}(\phi +\omega _s t;t) - \mathrm {c.c.}\right] \nonumber \\&\quad \times \left[ \mathrm {e}^{ i(\theta _{0y} - \omega _y s)} \sum _n F^\mathrm {coh}_{y\,n,z=0}(\phi +\omega _s s;s) - \mathrm {c.c.}\right] + {\mathcal {O}}(0)\nonumber \\&\quad \nonumber =\frac{\beta _y}{4 p_0^2}\left[ \mathrm {e}^{-i\omega _y(t-s)} \sum _m F^\mathrm {coh}_{y\,m,z=0}(0;t)\mathrm {e}^{-il_m(\phi +\omega _s t)} \sum _n F^{\mathrm {coh}*}_{y\,n,z=0}(0;s)\mathrm {e}^{il_n (\phi +\omega _s s)} + \mathrm {c.c.} \right] + {\mathcal {O}}(0)\nonumber \\&\quad =\frac{\beta _y}{4 p_0^2}\left[ \sum _m \sum _n \mathrm {e}^{-i(\omega _y+l_n\omega _s) u} F^\mathrm {coh}_{y\,m,z=0}(0;t) F^{\mathrm {coh}*}_{y\,n,z=0}(0;t-u) \mathrm {e}^{-i(l_m -l_n)(\phi +\omega _s t)} + \mathrm {c.c.} \right] + {\mathcal {O}}(0), \end{aligned}$$
(49)

where \( {\omega _y=\omega _y(J_x,J_y)=\omega _0 Q_y(J_x,J_y)} \) is the transverse frequencies of the individual particles, which depends on the transverse actions of the particles, the subscript \( z=0 \) in \( F^\mathrm {coh}_{y\,m,z=0} \) means that the headtail factor is canceled out, both \( \mathrm {c.c.} \) and the superscripted \( *\) imply complex conjugation, \( {\mathcal {O}}(0) \) refers to terms that later in the calculation would become 0, and the time parameter s has been exchanged for \( {u=t-s} \). In the first line, \( {\mathcal {O}}(0) \) refers to terms that later will become 0 due to the lack of noise in negative frequencies. In the second line, \( {\mathcal {O}}(0) \) additionally refers to non-resonant terms proportional to \( {\exp [i\omega _y(s+t)]} \) that will average to zero. For this reason, the factors \( \exp (i\theta _{0\,y}) \) cancel. In the second line, the dependence on the particles longitudinal phase \( {\phi +\omega _s t} \), including how it evolves with time, is also taken out of the coherent force.

To calculate the diffusion coefficient driven by noise and noise-excited wakefields, as modeled by the Vlasov theory derived in this paper, the last line in Eq. (49) must be treated as given by Eq. (46), with s exchanged for \( {u=t-s} \). Remember from Eq. (34) that \( \langle \cdot \rangle _z \) means an averaging over both \( \phi \) and \( J_z \). The averaging over \( \phi \) gives that \( {l_m=l_n} \), i.e., that modes close to different sidebands do not interact to drive a common diffusion since they are orthogonal. The calculation then becomes the double sum of the Fourier transform of the cross-correlation function of \( F^\mathrm {coh}_{y\,m} \) and \( F^\mathrm {coh}_{y\,n} \), which is the product of their Fourier transforms

$$\begin{aligned} D^{\mathrm {Vlasov}}_y(J_x,J_y) = \frac{\beta _y}{2p_0^2} \sum _m \sum _n \left\langle { {\hat{F}}^{\mathrm {coh}}_{y\,m}(\omega _y+l_m\omega _s) {\hat{F}}^{\mathrm {coh}*}_{y\,n}(\omega _y+l_n\omega _s) }\right\rangle _z, \end{aligned}$$
(50)

where a factor 2 has come from the inclusion of the complex conjugate in Eq. (49). Note that this expression is real, as it contains the sum over pairs of complex conjugated expressions. Looking at the expression for \( {{\hat{F}}^\mathrm {coh}_{y\,m}} \) in Eq. (41), we find that orthogonal modes with \( {\langle m_mm_n^*\rangle =0} \) will not contribute to the diffusion coefficient. In the unperturbed case, all modes are orthogonal. Since we have adopted the hypothesis of weak wakefields, we assume that the contribution of the terms with \({m\ne n}\) remains negligible compared to the diagonal terms. This hypothesis was confirmed numerically in all the cases studied. Therefore, we only keep the dominant terms with \( {m=n} \). The diffusion coefficient can finally be expressed as a function of the transverse actions as

$$\begin{aligned} D^{\mathrm {Vlasov}}_y(J_x,J_y)&= \sum _m D^{\mathrm {Vlasov}}_{y\,m}[\omega _y(J_x,J_y)] = \frac{\beta _y}{2p_0^2} \sum _m \left\langle {\left| {\hat{F}}^\mathrm {coh}_{y\,m}(\omega _y+l_m\omega _s)\right| ^2}\right\rangle _z, \end{aligned}$$
(51)
$$\begin{aligned} D^{\mathrm {Vlasov}}_{y\,m}(\omega _y)&= {\frac{\sum _i \left| {\hat{\xi }}_i(\omega _y+l_m\omega _s)\right| ^2 \left| \eta _{i\,m}\right| ^2}{2\beta _y}} \frac{\left| \varDelta \varOmega ^\mathrm {SD}_{l_m}(\omega _y+l_m\omega _s)\right| ^2}{ \left| \varDelta \varOmega _m-\varDelta \varOmega ^\mathrm {SD}_{l_m}(\omega _y+l_m\omega _s)\right| ^2} . \end{aligned}$$
(52)

The term \( {l_m \omega _s} \) in the argument to \( \varDelta \varOmega ^\mathrm {SD}_{l_m} \) will cancel the same term in Eq. (21). Hence, a mode with eigenfrequency \( \varOmega _m \) close to a sideband \( {\omega _0Q_{y0} + l_m\omega _s} \) can still drive diffusion of particles with betatron frequency \( {\omega _y\sim \omega _0Q_{y0}} \), which is far away from the sidebands. That is often the case in the LHC. Note that the noise signals \( \xi _i \) are given in units of the beam size, \( {\sigma _y=\sqrt{2J_y\beta _y}} \), so the factors \( \beta _y \) will cancel.

We will now compare the new Vlasov theory to the USHO theory derived in Ref. [3]. The USHO theory assumes that the motion of the unstable mode with complex frequency shift \( \varDelta \varOmega _m \) (with positive imaginary part) can be modeled as a single underdamped harmonic oscillator with complex frequency shift changed to \( \varDelta \varOmega ^\mathrm {LD}_m \) (with negative imaginary part) when stabilized by Landau damping. If \( {\mathrm {Im}\left\{ \varDelta \varOmega ^\mathrm {LD}_m\right\} >0} \), the Landau damping is not sufficient to stabilize the mode. The interpretation of \( \varDelta \varOmega ^\mathrm {LD}_m \) is further explained in Ref. [3]. Assuming small tune shifts, the MTF in the USHO theory is

$$\begin{aligned} \left. \frac{{\hat{\chi }}_m(\varOmega )}{\sum _i {\hat{\xi }}_i(\varOmega )\eta _{i\,m}}\right| ^\mathrm {USHO}= \frac{1/2}{\varDelta \varOmega ^\mathrm {LD}_m-\varDelta \omega _y }, \end{aligned}$$
(53)

where \( {\varDelta \omega _y=\omega _y-\omega _0Q_{y0}} \) is the frequency shift of the individual particles. Equation (53) can be compared to Eq. (40). All consequential differences originate in the different expressions for the MTFs. Assuming orthonormal mode shapes \( m_m \), the diffusion coefficient due to both noise and wakefields according to the USHO theory (only diffusion driven by wakefields was treated in Ref. [3]) can in the notation and framework used in this paper be written as

$$\begin{aligned} D^{\mathrm {USHO}}_{y\,m}(\omega _y) = \frac{\sum _i \left| {\hat{\xi }}_i(\omega _y + l_m \omega _s)\right| ^2 \left| \eta _{i\,m}\right| ^2}{2\beta _y} \frac{\left| \varDelta \varOmega ^\mathrm {LD}_m-\varDelta \omega _y-\varDelta \varOmega _m\right| ^2}{\left| \varDelta \varOmega ^\mathrm {LD}_m-\varDelta \omega _y\right| ^2}. \end{aligned}$$
(54)

Note that the noise was assumed dipolar and white in Ref. [3], giving \( {|{\hat{\xi }}_0|^2 = \sigma _{\xi 0}^2f_\mathrm {rev}} \) for \( {\omega \in [0,\omega _0/2)} \) and zero for other frequencies and noise signals. Here, \( \sigma _{\xi 0} \) is the rms amplitude of the noise kick per turn in units of the beam size. In addition, normalized units are used in Ref. [3], corresponding to \( {\beta _y=1} \). A factor \( \beta _y \) has therefore been added in the denominator of Eq. (54) to cancel that of the noise signal squared.

The derivation modeling the mode as an USHO made stronger assumptions on the dynamics and is therefore assumed to be less accurate than the new Vlasov-based theory. Even so, the only difference between the expressions is that the MTF in the Vlasov theory depends on the distance from the stability diagram to the undamped tune shift, \({{\varDelta \varOmega _m - \varDelta \varOmega ^\mathrm {SD}_{l_m}}} \), while the MTF in the USHO theory depends on the distance from the single-particle betatron frequency to the damped mode frequency, \( {\varDelta \varOmega ^\mathrm {LD}_m-\varDelta \omega _y} \). Separately, a great challenge with using the USHO theory, was the need to calculate \( \varDelta \varOmega ^\mathrm {LD}_m \), which in Ref. [3] introduced additional assumptions and inaccuracies through the use of a Taylor expansion. Hence, the MTF and diffusion coefficient derived in this paper are both assumed closer to reality and do not require further assumptions to be calculated.

4.2 Comparison to multi-particle tracking simulations

The expressions derived for the diffusion coefficients will here be benchmarked against multi-particle tracking simulation run with COMBI. Since diffusion is a second-order effect, more simulations are needed to get an accurate estimate of the diffusion coefficient that depends on the spread of action \( \langle \varDelta J^2\rangle \). 24 tracking simulations have been run with \( 10^{7} \) macro-particles for each configuration. Only horizontal rigid white noise \( {\propto \varXi _0} \) is modeled. The diffusion is presented in units of \( {D_0=\sigma _{\xi 0}^2/2\beta _x} \), the diffusion expected due to noise in the absence of wakefields and a feedback system.

First, the diffusion due to a single mode with \( {\eta _{0m}=1} \) is studied. The wakefields are in COMBI modeled by an anti-damper, i.e., a kick that is constant over the bunch length and that is proportional to the average position offset. This simplified model allows for an easy control of the corresponding tune shift in the numerical simulations. The numerical diffusion coefficient is in Fig. 3 compared to the Vlasov theory in Eq. (52) and the USHO theory in Eq. (54). In Fig. 3a, the tested mode has a tune shift qualitatively similar to that of the most problematic modes in the LHC, with a significant negative real part relative to the imaginary part. The two theories are close to identical and agree well with the simulations. In Fig. 3b, the tested mode has a tune shift such that the Taylor expansion used to get \( \varDelta Q^\mathrm {LD}_m \) in the USHO theory is inaccurate. The Vlasov theory is found to be superior to the USHO theory for this mode. For this setup with one rigid mode, the Vlasov theory in Eq. (52) is identical to its inspiration in Ref. [19] when using a two-dimensional tune spread.

Fig. 3
figure 3

Numerical diffusion calculated with COMBI compared to both the Vlasov and the USHO theories. The wakefields are driven by an anti-damper. In (a), the anti-damper represents a mode with complex tune shift \( {\varDelta Q^\mathrm {coh}_x=-1.467\times 10^{-3}+1.25\times 10^{-4}i}\). In (b), the anti-damper drives a mode with \( {\varDelta Q^\mathrm {coh}_x=3.75\times 10^{-4}+5.504\times 10^{-4}i} \). Both of these modes have a stability threshold, in terms of detuning coefficients, of \( {a_x=5\times 10^{-4}}\) and \( {b_x/a_x=-0.7} \). Note that what matters is the phase of the complex tune, not the amplitude, together with the relative stability margin in octupole current, which is \( {50\%} \) in both configurations. The red shaded area corresponds to 1 standard deviation from the solid red line

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Fig. 4
figure 4

Numerical diffusion calculated with COMBI compared to the Vlasov theory. The wakefields represent those in the LHC at top energy in 2018 with a linear chromaticity of \({Q'=15}\). The feedback gain is 0 in (a) and corresponds to a damping time of 100  turns in (b). The stability margin in octupole current is \( {25\%} \) in both configurations. The red shaded area corresponds to 1 standard deviation from the solid red line

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Second, the diffusion due to the full LHC wakefields will be studied. Calculations made with BimBim, using the impedance model for the LHC at top energy in 2018, have again been used to get the mode details. The bunch intensity, synchrotron tune, and rms particle momentum spread were the same as given in Table 1. The numerical diffusion coefficient is in Fig. 4 compared to the Vlasov theory in Eq. (51). There is good agreement both with and without a feedback. Note that the diffusion coefficient with a feedback is the sum of a multitude of \( D^\mathrm {Vlasov}_m \), driven by individual modes, which are shown individually in Fig. 5a. The sums for various stability margins are shown in Fig. 5b.

4.3 Distribution and stability evolution

Fig. 5
figure 5

Diffusion coefficient with the LHC wakefields, using the parameters in Table 1, according to the Vlasov theory in Eqs. (51) and (52). (a) displays the largest diffusion coefficients driven by individual wakefield-driven modes with a stability margin of \( {25\%} \). The stability margin is a global measure, being the margin for the least stable mode, which corresponds to the most peaked line in (a). (b) displays the total diffusion coefficient for various stability margins. The green curve with \( {25\%} \) stability margin is equal to the blue curve in Fig. 4b and the sum of the curves in (a)

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The diffusion driven by noise and wakefields causes an evolution of the distribution, which leads to a change of the stability diagram since the latter is inversely proportional to the dispersion integral in Eq. (21). We consider here an illustrative example for the horizontal plane in the LHC at top energy in 2018 that is also considered in Ref. [22], with most parameters given in Table 1, a ratio between the detuning coefficients of \( {b_x/a_x=-0.7} \), and a detuning coefficient \( a_x \) that is \( {25\%} \) larger than the initial stability threshold for a bunch with a Gaussian transverse distribution. This is the same case as is tested in Fig. 4b. The diffusion coefficient is a sum of a multitude of \( D^\mathrm {Vlasov}_m \) that are shown individually in Fig. 5a. The modes that are inherently stable also without Landau damping, in part due to the feedback, are bowl-shaped, i.e., zero at a center tune and increasing on both sides. This is qualitatively the same shape that is found in Ref. [23] for a bunch driven by noise and damped by a feedback system, in the absence of wakefields. It is found in Ref. [23] that diffusion coefficients of this shape are not critical for the loss of Landau damping that is observed in the LHC. On the other hand, the modes that would be unstable if not for Landau damping are peaked and narrow as the ones in Fig. 3. It is found in Ref. [3] that diffusion coefficients of this shape lead to a local flattening of the distribution in action space and a drilling of a hole in the stability diagram until the mode is unstable. This is the cause for loss of Landau damping after a latency that has been observed in the LHC.

Table 1 Parameters for an LHC test case
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Fig. 6
figure 6

PyRADISE simulations with the LHC wakefields, using the parameters in Table 1 and a stability margin of \( {25\%} \). Relative change of the distribution on the left (a and c). Evolution of the stability diagram on the right (b and d), with the tune shifts \( \varDelta Q^\mathrm {coh}_m \) of the modes marked by black crosses. Only the least stable mode is included on top (a and b), while all the modes are included on the bottom (c and d). The drilling of the hole in the stability diagram is only visualized until the initial stability margin is reduced by \( {75\%} \), because of numerical effects discussed in Sect. 4.5

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The distribution and stability evolution according to the Vlasov theory, both driven by the diffusion of only the least stable mode and by the diffusion of all the modes, are presented in Fig. 6. These evolutions have been calculated using the code PyRADISE [3], using a \( {2000\times 2000} \) grid in transverse action space, equidistant in the transverse amplitudes \( {r\propto \sqrt{J}} \). The latencies found in both simulations are gathered in Table 2, which also include the latencies found with the USHO theory numerically and analytically. The relative change of the distribution driven by only the least stable mode in Fig. 6a appears quite different from that driven by all the modes in Fig. 6c. However, what matters most is the local flattening of the distribution that is visible in both figures as a line across which particles have moved from the dark blue area on the left to the dark red area on the right. The stability diagram therefore evolves similarly due to only the least stable mode in Fig. 6b as due to all the modes in Fig. 6d. Since the additional modes drive a measurable diffusion that in this case partly counteracts the distribution evolution driven by the least stable mode, their inclusion increases the latency from \( 24\hbox { s}\) to \( 37\hbox { s}\). This increase by about \( {50\%} \) is considered small on the vast scale of possible latencies [3], as will be exemplified in Sect. 4.4.

Table 2 Latencies found with an anlytical equation and with several numerical simulations of different models [22]
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4.4 Dependence of latency on the stability margin

The latency depends on multiple key parameters, such as the noise and the noise-mode moments, but on none as critically as the stability margin, which is how much more detuning is used than necessary to stabilize a Gaussian beam. Here, this dependence will be illustrated by calculating the latency for the same test case as in Sect. 4.3, but with different stability margins.

The latencies calculated with the same five approaches as in Table 2 are shown in Fig. 7. There are many key results. First of all, the latency increases by more than 4 orders of magnitude by increasing the stability margin by 1 order of magnitude. Secondly, the analytical latency agrees well with the simplified simulation based on the USHO theory for all stability margins. This is expected, as the two approaches make the same assumptions. In general, the analytical latency gives an acceptable estimate also compared to the less simplified simulations, but it is slightly longer. The Vlasov and USHO theories with a single mode agree well, especially at small margins. Note that the Vlasov theory with a single mode has a numerical challenge at large stability margins that will be discussed in Sect. 4.5.

The model most true to the actual physics are the simulations based on the Vlasov theory including all the modes. This is also the most complex mechanism. The initial diffusion coefficient for different stability margins is displayed in Fig. 5b. At low margins, the peak due to the least stable mode is clearly visible, and it is this mode that eventually goes unstable. However, the diffusion due to the other modes slows down the destabilizing process. The latency with the Vlasov theory with all the modes is longer than the latency with only the least stable mode, and the difference increases with the stability margin, for small margins. Then, starting at a margin of \( {50\%} \) the peak due to the least stable mode is no longer visible in Fig. 5b and the total diffusion drills a wider hole. Eventually, it is actually a different mode that goes unstable, which is centered more in the middle of the peak of the total diffusion coefficient. This transition can also be seen in Fig. 7.

Fig. 7
figure 7

Latency as a function of the stability margin for the LHC case otherwise described by the parameters in Table 1. The different curves correspond to different approaches (the same as in Table 2) to calculate the latency

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4.5 Discussion

Several aspects of this mechanism deserve further discussion. Here, we will discuss the physics, the differences between the USHO and the Vlasov theories, and various numerical challenges with both.

It is only the Vlasov theory that can accurately model the full wake-driven diffusion. The USHO theory can model the impact of additional almost unstable modes, but not the stable ones that cause the bowl-shaped diffusion coefficients exemplified in Fig. 5a. As also discussed in Ref. [3], it is challenging to predict the impact of the secondary modes in addition to the least stable one. In some cases, as the one tested here, they will extend the latency, but they may in principle also shorten it. If this level of understanding is required, it must be evaluated on a case-by-case basis.

The Vlasov theory for the MTF has been derived in this paper. It is illustrated in Fig. 3 that the Vlasov theory is more accurate than the more simplified USHO theory derived in Ref. [3]. However, the fact that the USHO theory allows for the derivation of an analytical equation for the latency (Eq. (50) in Ref. [3]) is quite useful. As shown in Fig. 7, the analytical estimate is in general within short range of the simulated latencies, in comparison with the many orders of magnitude over which the latency varies with the stability margin. Therefore, we suggest to use the analytical latency equation to estimate how much stability margin is required to achieve a desired latency. The required margin depends in particular on the noise and wakefield eigenmodes.

An aspect, which has not been focused on here, is the fact that the diffusion modeled by both theories is the expected motion as the number of turns goes to infinity. As seen by the variability between the separate simulations in Figs. 3 and 4, the expected diffusion coefficient is clearly not achieved each turn. This was discussed in more detail in Sect. IV.C in Ref. [3]. In fact, it was found that the wakefields driven by a mode particularly close to the stability limit drove a single-particle motion more comparable to resonant motion than diffusion.

A numerical issue has been discovered with the Vlasov theory. Loops start to develop at the edge of the drilled hole in the stability diagram when there is a large initial stability margin, especially when it models only a single mode. It is visible already in Fig. 6. These loops continue to grow so much that it interrupts the drilling of the initial hole, in addition to making the physical interpretation of the stability diagram challenging. We have found that this behavior is caused by initially insignificant details in the diffusion coefficient, which begin to enhance themselves. Based on the discussion in the previous paragraph, about the diffusion being the expected distribution evolution and not the actual evolution per turn, we believe this enhancement of minor details to be a non-physical consequence of the theory. Furthermore, this behavior is not reproduced with the USHO theory. For this reason, the latency is found by extrapolating the drilling from when the initial stability margin is reduced by \( {75\%} \).

A final numerical challenge that we would like to point out is the convergence with numerical parameters. By reducing the discretization step in time and action space, the latency does converge towards a single value, as expected. Similarly, the stability diagram is calculated as the one-sided limit when the small parameter \( {\epsilon \rightarrow 0^{+}} \) [6]. However, for \( {\epsilon \rightarrow 0^{-}} \) and \( {\epsilon =0} \), one gets different results. In other words, the dispersion integral in Eq. (21) is discontinuous in the imaginary part of \( \varOmega \), which causes numerical challenges for small positive values of \( \epsilon \). In the PyRADISE calculations presented here, we have used \( {\epsilon =\left| a_x\right| /200} \), where \( a_x \) is the in-plane detuning coefficient due to octupole magnets. With this value of \( \epsilon \), the latency and stability diagram calculation appear to have converged, but it is quite close to a threshold below which the results are inaccurate due to the discontinuity of the dispersion integral.

5 Conclusion

In this paper, we have studied the excitation of beams in high-energy synchrotrons by noise. Within the framework of the linearized Vlasov equation, the MTF has been derived, modeling how a wakefield beam eigenmode is excited by noise. The initial excitation by noise depends on the shape of the eigenmode and the correlation of the noise along the bunch, modeled through the noise-mode moments \( {\eta _{i\,m}} \). The future evolution of the mode, after being excited by noise, depends on the strength of the wakefields and the details of the detuning. It has here been assumed that the wakefields are weak, so that modes of different azimuthal mode numbers are linearly independent.

The BTF including chromaticity and wakefields has been derived based on the MTF. By comparing to multi-particle tracking simulations, it has been found that the new semi-analytical expression accurately models the impact of wakefields on the BTF. In particular, the shift of the BTF amplitude peaks in tune space has been thoroughly explained. The new theory can, in the limit of negligible wakefields, also be used to get an expression for the BTF including chromaticity. This expression also agrees excellently with multi-particle tracking simulations.

The main motivation for this work was to better understand the loss of Landau damping driven by noise and wakefields, which has been observed in the LHC. The MTF based on the linear Vlasov equation has been used to derive an expression for the diffusion driven by the interplay between noise and wakefields. The Vlasov-based theory can both model the diffusion driven by almost unstable modes, for which the diffusion coefficients are peaked in tune space, and the diffusion driven by inherently stable modes, for which the diffusion coefficients are bowl shaped in tune space. The peaked diffusion coefficients driven by the almost unstable modes can cause a local flattening of the distribution, leading to the drilling of a hole in the stability diagram. This mechanism can cause a loss of Landau damping. By comparing to multi-particle tracking simulations, the Vlasov theory is found superior to the previously derived USHO theory, which modeled the eigenmodes as underdamped stochastic harmonic oscillators. Nevertheless, by use of the PDE-solver PyRADISE, both theories produce similar latencies within about a factor 2, which is fairly close compared to the many orders of magnitude that the latency varies over. The main technique suggested to increase the latency and thereby mitigate the loss of Landau damping is to increase the stability margin, i.e., operating with more detuning than predicted to barely stabilize a beam with a Gaussian transverse distribution. How much margin that is needed to achieve a certain latency depends in particular on the noise and wakefield eigenmodes and can be estimated with the analytical latency equation derived in Ref. [3].

Going forward, it will be of particular interest to study operational conditions of real accelerators with the new diffusion theory for the loss of Landau damping. For instance, the impact of crab cavity amplitude noise must be studied in detail to estimate the accepted noise levels in the HL-LHC. Such a study may also find alternative operational configurations, by e.g., optimizing the linear chromaticity and gain of the transverse feedback system so that the latency is maximized.